Length of dislocations

A mesoseale variable deseribing a eolleetion of atomie disloeations is the total length of dislocation line per unit volume in metallurgieal publieations... 

Dislocation density is measured as the total length of dislocation lines in a unit volume of crystal, meters per meter cubed. However, experimentally it is often simpler to determine the number of dislocations that intersect a surface, so that a common measure of dislocation density is the number of dislocation lines threading a surface, that is, the number per meter squared. In a fairly typical material there will be on the order of 108 dislocation lines crossing every square centimeter of solid. However, it is known that if a solid is deformed, the dislocation density rises, perhaps by a factor of 103 or 104. Clearly, dislocations must be able to multiply under the conditions that lead to deformation. 

Some defects have extent in only one dimension—line defects. The most prominent of these, the dislocation, is a line in the crystal along which atoms have either an incorrect number of neighbors or neighbors which have not die correct distance or angle. In 1 cubic centimeter of a real crystal, one might find a wide variation of length of dislocations present—from near zero to perhaps 1011 centimeters. 

Dislocation density is often determined by counting the number of dislocations per area intersecting a polished surface. If the dislocation density in cold-worked copper is found to be 2 x 1010/cm2, what is the total length of dislocation line per volume ... 

The physical interpretation of this result is that it is the force per unit length of dislocation acting on a segment by virtue of the presence of the stress field a. This expression is the famed Peach-Koehler formula and will be seen to yield a host of interesting insights into the interactions between dislocations. 

The deformation is believed to occur mainly by climb of dislocations, with oxygen atoms traveling along dislocation loops. Lengths of dislocation loops vary greatly but are of the order of magnitude of 1 pm. 

The number of dislocations in a unit volume of crystal is the dislocation density. Since it is taken to be the total length of dislocations per unit volume, its units are cm"2. A less precise, but more readily measurable, quantity is the number of dislocations intersecting the unit area of the crystal, and this is also expressed in miits of cm 2. For isotropic materials the two methods of defining dislocation density are almost equivalent (to within a factor of 2). But for anisotropic materials, e.g., layer structures, the two methods may yield vastly different values (factors of up to 10 ). 

Performing the integration and assuming the logarithmic term can be represented by a geometric coefficient a, one obtains an energy per unit length of dislocation... 

Plastic deformation results from the accumulated motion of numerous dislocations at the atomic scale. The dislocation density p is a parameter representing the average amount of accumulated plastic deformation or, in other words, the amount of deviation from the strictly geometrical lattice structure. The dislocation density p is defined as the total length of dislocation lines per cubic centimeter, and it is almost identical to the flow stress of a metal under hot forming or the internal stress Oi. Numerous dislocations are introduced by forming. For example, the initial dislocation density of a fully annealed structure po is about 10 (cm/cm ) but increases to lO (cm/cm ) or more after metal forming. 

The expressions for the energies above were given per unit length of dislocation. When not normalized for a unit length, this energy is proportional to the length of the dislocation ... 

If one defines the force on a unit length of dislocation as the work done when the unit length of the dislocation segment, dl, moves a unit distance, dx , one gets ... 

The HP relation may be further discussed as follows. A stress concentration exists at or close to the obstacle (a grain boundary in our case). The key factor in the motion of dislocations is the first or leading dislocation in the vicinity of the obstacle. Assume that the leading dislocation has moved a distance, dx all the trailing dislocations will move the same distance. The work done per unit length of dislocation (in Chap. 3) is ... 

Let there be several obstacles in our material with a distance of 2A between them (figure 6.23). Consider a dislocation pinned on these obstacles. When the external stress r acts on the dislocation, it tries to move on and bows out. Its shape is a segment of a circle because this covers the greatest area with the least-most energy to create new length of dislocation line. 

Let us consider further the free path length of dislocations, As it is known for metals [3], in which the main role in plastic deformation belongs to the mobile dislocations, assesses as 10 A. For polymers, this parameter can be estimated as follows [28] ... 

The values for different polymers, A assessed by the Eq. (4.6) is about 2.5 A. The same distance, which a segment passes at shearing, when it occupies the position, shown in Fig. 4.1 b, that can be simply calculated from purely geometrical considerations. Hence, this assessment also indicates no reasons for assuming any sufficient free path length of dislocations in polymers rather than transition of a segment (or several segments) of macromolecule from one quasiequilibrium state to another [31]. 

Defects as shown in Fig. 1(a) are the subject of the present chapter. They are referred to as metadislocations, and occur in numerous structurally complex metallic materials. The concept of metadislocations addresses a central problem in the plasticity of materials with large lattice parameters In these materials, conventional dislocation-based deformation mechanisms are prone to failure. This is a direct consequence of the elastic strain energy, which, per unit length of dislocation, is given by... 

The elastic stress field around a dislocation affects the elastic energy of a dislocation and the interaction between parallel dislocations. The elastic energy per unit length of dislocation between two cylindrical surfaces of radius Cq and R is given by ... 

The strain energy associated with edge and screw dislocations may be determined from Eqs. (4a)-(4e) and (5). The results for the strain energy per unit length of dislocation are " ... 

When a sufficiently large external force is applied to a crystalline material containing dislocations the dislocations move, accomplishing plastic deformation, as previously discussed (see Figure 6). If an external stress is applied to the crystal, then forces are applied to the dislocations which it contains. The force on a dislocation per unit length of dislocation is ... 

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